Tetrachloroethene (PCE, 99% purity, ACS grade) and cis-1,2-dichlorotethene (cis-DCE, 97% purity) were purchased from Sigma-Aldrich Inc. Acetonitrile (99.9% purity, ACS high-performance liquid chromatography (HPLC) grade), dimethyl sulfoxide (DMSO, 99.9% purity), methanol (99.9% purity, ACS HPLC grade), sodium bromide (NaBr, 99.9% purity, ACS grade), and sodium chloride (NaCl, 100% purity, ACS grade) were obtained from Fisher Scientific. Isopropanol (99.9% purity, HPLC grade) was obtained from Acros Organics. All chemicals were used as received. The water used in all batch and column experiments was purified to a resistivity > 18.1 MΩ×cm and total organic carbon (TOC) < 100 ppb using a Milli-Q ultrapure water system (model Gradient A10, Millipore).
 Ottawa sands (U.S. Silica Co.) were used to construct porous media employed in the column experiments. ASTM 20–30 mesh sand was used as received, while ASTM 45–50 and ASTM 50–60 mesh sands were produced by sieving Federal Fine sand (ASTM 30–140 mesh). The mean grain size diameter, d50, for the 20–30, 45–50, and 50–60 mesh sands are 725, 328, and 275 m, respectively. Uniformity indices, Ui, for these fractions are 1.20, 1.09, and 1.10, respectively.
2.2. Batch Methods
 Liquid-liquid equilibrium experiments were conducted in 35 mL glass centrifuge tubes sealed with Teflon-lined screw-on caps following methods similar to those reported by Ramsburg et al. . In brief, tubes were loaded with Milli-Q water, cis-DCE, and PCE to arbitrarily predetermined overall mole fractions and allowed to mix on LabQuake oscillating shaker trays at 22.0 ± 0.1 °C for 72 h. Previous experiments have demonstrated 72 h is sufficient time for liquid phase equilibration [Cope and Hughes, 2001, Ramsburg and Pennell, 2002]. Following the equilibration period, the batch systems were centrifuged at 1500 rpm for 10 min to consolidate each phase prior to separation and sampling. Approximately 5 g of the aqueous phase and 10 L of the organic phase were diluted (by mass) into a mixture of Milli-Q water and DMSO to fall within the linear response range of the gas chromatographic method (see analytical methods). DMSO was employed as a cosolvent at approximately 11% by weight in all diluted samples.
2.3. Column Methods
 Kontes borosilicate glass columns (4.8 cm inner diameter (ID)) were used for all transport experiments. Each column was dry packed with an Ottawa sand fraction (described in section 2.1), flushed with CO2 for a minimum of 10 pore volumes, and then saturated by flushing Milli-Q water through the column for 20 pore volumes. The PCE used as the DNAPL was dyed red with 1 × 10−4 M of Oil Red O (Alfa Aesar) for visualization within the columns. PCE-DNAPL entrapment was performed following established methods [Pennell et al., 1993]. Nonreactive, conservative tracer tests were conducted using aqueous solutions of 0.01 M Br− before and after the introduction of DNAPL to characterize the porous medium employed in each column experiment. Subsequent to the completion of the second tracer test, a pulse (approximately 250 min) of aqueous solution containing approximately 165 mg/L cis-DCE was introduced to the column at a rate of 2.0 mL/min using an Agilent Series 1100 quaternary pump. The cis-DCE pulses were produced from the column using Milli-Q water delivered at the same rate. The concentrations of cis-DCE and PCE in the column effluent were quantified using an inline sampling and analysis system described in section 2.4. Following the cessation of flow, the column was deconstructed and extracted for mass balance purposes.
2.4. Analytical Methods
 Batch experiment samples were quantified in triplicate using a Hewlett-Packard 5890 Series II gas chromatograph (GC) equipped with a Perkin Elmer TurboMatrix 40 Trap headspace sampler and flame ionization detector. The GC was calibrated prior to each day of use with a seven-point calibration curve. Headspace vials containing the diluted samples were heated to 95 °C and shaken for 25 min prior to sample transfer to the GC where separation was accomplished on a HP-Plot Q capillary column (30 m length, 0.53 mm ID, 40 m film thickness).
 Column experiments employed an inline sampling system comprising a 10-port, 2-position electric valve (model EHMA, Valco Instruments Co. Inc.) fitted with a 20 L sample loop. The valve was plumbed in line with: (1) the effluent end of the column; and (2) an Agilent Series 1100 HPLC. The HPLC was equipped with an Agilent Zorbax Eclipse XDB-C8 packed chromatography column (150 mm length, 4.6 mm ID, 5 m particle diameter) maintained at 40.0 °C and a diode array detector set to quantify absorbance at 254 nm. The HPLC was operated at 1 mL/min under isocratic conditions with mobile phase composition of 80% acetonitrile and 20% Milli-Q water. Under these conditions, the average HPLC column retention times for cis-DCE and PCE were approximately 2.3 and 3.2 min, respectively. The HPLC system was calibrated before each day of use using a 6-point standard curve established by introducing standards through a 20 L sampling valve. The inline system was programmed to sample the column effluent every 4.0 min. At the end of each experiment the contents of the column were extracted with isopropanol to close mass balances. Extract samples were diluted to fall within the range of a 6-point calibration curve, and analyzed using an Agilent Series 1100 autosampler connected to the HPLC. The HPLC method was similar to that employed for effluent sampling but excluded commands for firing the Valco valve.
 Effluent samples collected during nonreactive tracer tests were analyzed for bromide and chloride using a Dionex ICS-2000 Ion Chromatograph (IC) equipped with an AS50 autosampler and fitted with an IonPac AS18 anionic-exchange column (250 mm length, 4 mm ID, 7.5 m particle diameter). Isocratic separation was accomplished at 1 mL/min using a mobile phase containing 23.0 mM KOH. Under these conditions, the average retention times for Cl− and Br− were approximately 4.4 and 6.8 min, respectively. The IC was calibrated before each day of use using a 6-point standard curve with calibration checks performed every 20 samples.
 The densities of the liquid phases present in the batch experiments were quantified in triplicate at 22.0 ± 0.1 °C using 2 mL glass pycnometers (Ace Glass). Each day of use the pycnometers were calibrated using Milli-Q water. Calibrations were subsequently verified using isopropanol. Water content was quantified using a Karl Fisher titrator (model DL38, Mettler Toledo). The instrument was calibrated with a water standard (1% water by weight, EMD Chemicals, Inc.) on each day of use.
2.5. Thermodynamic Modeling
 Ternary phase behavior was predicted using an isothermal flash calculation [Smith and Van Ness, 1987] performed in Matlab version R2009b (The MathWorks, Inc.) with UNIFAC [Fredenslund et al., 1975] employed to estimate activity coefficients. Iteration convergence was established when successive changes to molar phase fractions and component mole fractions were less than 1 × 10−10. No attempt was made to adjust the UNIFAC parameters to fit the data. Structural parameters for the relevant subgroups were obtained from the work of Hansen et al.  with water assigned as 1 × H20 (group 7), cis-DCE assigned as 1 × HC=C (group 2) and 2 × Cl-C=C (group 37), and PCE assigned as 1 × C=C (group 2) & 3 × Cl-C=C (group 37). Interaction parameters between groups m and n are am,n and an,m, both defined in Kelvin, where: am,m = an,n = 0 (by definition); a2,7 = 785.6, a7,2 = −26.52 [Cooling et al., 1992]; a7,37 = 651.9, a37,7 = 1100 [Cooling et al., 1992]; and a2,37 = 237.3 and a37,2 = −3.167 [Gmehling et al., 1982].
2.6. Mathematical Modeling
 Simulation of contaminant transport in the NAPL-contaminated columns was based on the solution of component mass balance equations assuming steady flow and an immobile (entrapped) NAPL saturation [Abriola, 1989]. Component mass balance equations solve for the spatial-temporal distribution of a mass concentration of component i within each bulk fluid phase (aqueous: ; organic: ):
where is the matrix porosity, is the -phase saturation, is the three-dimensional hydrodynamic dispersion tensor for component i in phase [Bear, 1972], is the -phase pore velocity computed using a modified form of Darcy's law [Abriola, 1989], and is the interphase mass exchange of component i from the - to the -phase. In this work, interphase mass exchange is assumed to occur only between the aqueous and organic phases (i.e., sorption to soil particles is neglected) and is approximated using a linear driving force expression [Weber and DiGiano, 1996]:
where is the equilibrium solubility of component i in the -phase, and is a lumped mass transfer coefficient, which quantifies the rate of mass transfer of component i from the - to the -phase. This lumped coefficient can be obtained by employing a Sherwood number correlation [e.g., Miller et al., 1990; Imhoff et al., 1994; Powers et al., 1992, 1994a] where d50 is the mean grain diameter, and Dami is the aqueous phase molecular diffusion coefficient of component i. Dissolution requires use of the specific interfacial area to account for the transient nature of mass transfer resulting from reductions in interfacial area. The specific interfacial area (as) of the organic phase globule was approximated with a uniform distribution of spherical globules, such that as = 3/b, where b is the NAPL globule radius. Simulations of the column experiments reported here, however, assume a pseudo-steady flow with negligible changes to NAPL volume or specific interfacial area resulting from absorption and dissolution.
 Writing equation (1) for the aqueous and organic phase results in two transport equations for each component i, one describing transport in the aqueous phase and one describing transport in the organic phase:
 Recognizing that mass balance constraints require equations (3a) and (3b) can be combined to form a single equation for component i across both the aqueous and organic phases:
 Assuming pseudo-steady state conditions and spatially uniform saturation in one-dimensional (1-D) flow, equation (4) simplifies to the 1-D advection-dispersion equation (ADE) commonly employed in the sorption literature:
where, for NAPL partitioning, and all other parameters are as given previously.
 Equation (5) has been solved for a variety of interphase mass exchange conditions. Among the simplest and most commonly employed is the local equilibrium condition where concentrations of component i in the aqueous and organic phases are related algebraically using an equilibrium partitioning coefficient defined in terms of concentration :
 Employing equation (6), equation (5) simplifies to an ADE assuming equilibrium partitioning (ADEQ):
where R represents the retardation factor of the contaminant, assuming equilibrium exchange with the immobile organic phase: . Equation (7) is commonly employed in the sorption and PITT literature and has been solved for a variety of initial and boundary conditions [e.g., van Genuchten and Alves, 1982].
 Under conditions of kinetic interphase mass exchange, equations (2) and (3) are combined and simplified by assuming exchange is proportional to the concentration gradient between the contaminant concentration at the organic phase surface and in the bulk aqueous phase (Cia):
 Equation (8), abbreviated here as the ADEK (ADE with kinetic partitioning), has again been simplified to 1-D pseudo-steady state flow and makes use of the equilibrium relation provided in equation (6) to quantify the equilibrium contaminant concentration in the organic phase. When coupled with appropriately applied initial and boundary conditions, equation (8) can be solved analytically or numerically. Models based on equation (8) are commonly referred to in the sorption literature as one-site kinetic adsorption models [e.g., van Genuchten and Wagenet, 1989; Leij et al., 1993; Toride et al., 1993]. A popular modeling program, CXTFIT, employs this model for quantifying sorption parameters [Toride et al., 1999]. Note that equation (8), however, assumes that the organic phase is well mixed, and therefore does not consider the potential effect of concentration gradients present within the NAPL globule. While appropriate for certain situations, this may not always be the case, resulting in the need to model the diffusion of contaminant in the immobile organic phase.
 Rasmuson and Neretnieks  used the work of Pellett  to derive an often-employed analytical solution to equation (5) when diffusion within the immobile phase is important. In their solution, equation (5), which is derived assuming volume-averaged concentrations (Cia and Cin), is coupled to a radial diffusion equation written for the contaminant concentration (Cin*) within the organic phase using the interphase mass exchange term as a boundary condition (equation (9b)) applied at the surface of the immobile organic phase (r = b):
 Dnmi in equation (9) is the organic phase molecular diffusion coefficient of component i, r is the radial dimension, and b is the NAPL globule radius. Equation (9), abbreviated here as ADEKD (ADEK with diffusion), models the transport of contaminant in the aqueous phase undergoing partitioning to the immobile organic phase, where interphase mass exchange is controlled by both (1) resistance in the aqueous phase boundary layer separating the aqueous and organic phase (equation (9b)) and (2) resistance to mass exchange because of diffusion within the immobile organic phase (equation(9c)). Although originally derived to model adsorption because of partitioning into immobile regions of soil grains [e.g., Rasmuson and Neretnieks, 1981; van Genuchten, 1985; Crittenden et al., 1986], the equations are extended in this work to model partitioning into an immobile organic phase. Table 1 summarizes the parameters employed in the diffusion model and compares the nomenclature between the sorption literature and the liquid-liquid partitioning application of this work.
Table 1. Parameter Equivalence Between Solid-Liquid Sorption Model and Liquid-Liquid Partitioning Model
|b (L)||Soil particle radius||b (L)||NAPL globule radius|
|Ds (L2/T)||Intra-aggregate diffusivity||Dnmi (L2/T)||Diffusivity in NAPL globule|
|K (L3 L−3)||Volume equilibrium constant||Kpi (L3 L−3)||Equilibrium partitioning coefficient|
|kf (L/T)||Mass transfer coefficient|| (L/T)||Mass transfer coefficient|
|m (-)||m (-)||sa/sn|
| (M/L3)||Vol. avg. contaminant concentration in particles||Cin (M/L3)||Vol. avg. contaminant concentration in NAPL globule|
|qi (M/L3)||Internal contaminant concentration in particles||Cin* (M/L3)||Internal contaminant concentration in NAPL globule|
|qs (M/L3)||Internal contaminant concentration at particle surface (qi(b, z,t))||Internal contaminant concentration at NAPL surface (Cin*(b,z,t))|
|r (L)||Radial distance from center of spherical soil particle||r (L)||Radial distance from center of spherical NAPL globule|
| (L3 L−3)||Soil porosity||n (L3 L−3)||Soil porosity|
 The described models were employed here to simulate cis-DCE breakthrough curves from columns contaminated with residual PCE-DNAPL. The Brenner  solution to equation (7) published by van Genuchten and Alves  for a finite-length column was fit to tracer tests conducted before and after the establishment of the PCE-NAPL entrapped saturation. Pre-NAPL tracer results were employed to obtain best-fit soil column porosities , while post-NAPL tracer results were fit to determine the longitudinal dispersivity and NAPL saturation (Sn) in each column. Gravimetrically determined values for porosity and NAPL saturation were not explicitly used; rather the experimental values guided the model fits, which were assumed to provide a better approximation of the actual column conditions. The exception here was Column A, where gravimetrically determined PCE-NAPL saturation was employed because tracer fitting produced an unreasonably high saturation (> 20%). Table 2 summarizes the column parameters for all three columns employed here. All parameters other than and Sn were independently determined through measurement or constitutive relationships available in the literature. Note that the NAPL globule radius (b = deff,blob/2) employed here is based on a newly formed correlation using a comprehensive survey of the literature, as described in section 3.2.
Table 2. Column Parameters
|Sand size (ASTM sieve sizes)||50–60||45–50||20–30|
|Range of grain diameter (m)||250–300||300–355||600–850|
|Median grain size diameter (d50) (m)||275||328||725|
|Uniformity index (Ui)||1.10||1.09||1.20|
|Column diameter (dc) (cm)||4.80||4.80||4.80|
|Column length (L) (cm)||3.50||3.45||3.50|
|Porosity (n) (Lvoid/Ltotal)||0.388||0.377||0.360|
|Pore volume (PV) (mL)||24.6||23.5||22.8|
|PCE-NAPL saturation (Sn) (%)||17.1||13.2||11.1|
|PCE-NAPL volume (Vn) (mL)||4.20||3.10||2.53|
|PCE-NAPL ganglia radius (b)a (m)||128||141||263|
|Volumetric flow rate (Q) (mL/min)||2.00||2.00||2.00|
|Darcy velocity (q) (cm/sec)||1.84 × 10−3||1.84 × 10−3||1.84 × 10−3|
|Pore water velocity (v) (cm/s)||5.72 × 10−3||5.63 × 10−3||5.75 × 10−3|
|Longitudinal dispersivity (cm)||1.76 × 10−1||8.13 × 10−2||3.37 × 10−1|
|Diffusivity of cis-DCE in the aqueous phase (Da)b (cm2/sec)||1.07 × 10−5||1.07 × 10−5||1.07 × 10−5|
|Diffusivity of cis-DCE in PCE-NAPL (Dn)b (cm2/sec)||1.21 × 10−5||1.21 × 10−5||1.21 × 10−5|
|cis-DCE partition coefficient (Kp) (Laq/LNAPL)||105||105||105|
|Retardation factor (equilibrium partitioning) (Rc) (-)||22.6||16.9||14.1|
|Modified Sherwood number (Sh′) (-)||0.246||0.305||0.856|
|Lumped mass transfer coefficient (kl) (s−1)||3.49 × 10−3||3.05 × 10−3||1.75 × 10−3|
|cis-DCE pulse concentration (C0) (mg/L)||166||167||166|
|Pulse duration (t0) (min)||208.90||208.60||205.34|
|Experiment duration (tf) (min)||652.21||656.43||654.44|
|cis-DCE recovery in effluent (%)||101.3||101.0||99.2|
|cis-DCE mass balancec (%)||102.0||101.2||101.0|
 Following parameter estimation from the tracer results, a solution to a nondimensionalized form of equation (9) published by Rasmuson and Neretnieks  was employed to predict cis-DCE breakthrough. The Rasmuson and Neretnieks solution is described here since this use is the first application for estimating contaminant transport in a NAPL-contaminated region. Rasmuson and Neretnieks employed Laplace transforms to solve equation (9) subject to the following initial and boundary conditions:
 The solution, written in nondimensional form, is:
with model parameters described in Table 3. Equation (11) provides an efficient, analytical solution that can be applied to solute partitioning in NAPL source zones.
Table 3. Nomenclature Employed in ADEKD Solution (Equation (11))
| (L/T)|| |
|b (L) Dnmi (L2/T) KpCi (L3 L−3)|
| || |
|m (-)|| || |
|V (L/T) Dia(L2/T) z (L)|| || |
 Equation (11) was implemented in Matlab version R2010a. Numerical integration of the infinite integral followed the methods of Rasmuson and Neretnieks  in which the infinite integral is replaced by a finite integral with chosen as a sufficiently large upper bound. Proper model implementation was verified by comparison with other analytical [van Genuchten and Alves, 1982] and numerical solutions [Toride et al., 1999]. It is worth noting that equation (11) can be made to approximate solutions to the simpler partitioning formulations (equations (7) and (8)). To approximate equilibrium sorption using equation (11), and Dnmi are assumed to be large ( >> 10−1 s−1 and Dnmi >> 10−4 cm2/s), such that partitioning across the aqueous phase boundary layer is nearly instantaneous and diffusion within the NAPL is rapid. If rate limited mass transfer is expected, but diffusive limitations within the NAPL are negligible, equation (11) is simplified to the kinetic equation (equation (8)) by assuming Dnmi is large. Global mass balance calculations confirmed that errors because of numerical integration were never more than 1%.